# Full text of "Treatise On Analysis Vol-Ii"

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```212       XII!    INTEGRATION

Assume that

inf f(x) ^(U */)0>) f£ sup f(x)

for all y e Y and all/e ^R(X). Show that under these conditions there exists a vaguely
continuous mapping a: y\-~*ay of Y into M+(X) such that ay is carried by Tr~"l(.y) for
all y e Y, and such that (U -/)(7r(.x)) = </, aw(X)> for all x e X, Consider the converse.
(Consider the transpose '£/: MR(Y)^MR(X) of U (12.15.3).)

9. Let X be a compact space, let p. be a diffuse positive measure on X such that p,(X) = 1,
and let u: X->X be a bijection such that both u and w"1 are /x-measurable, p, is u-
invariant (Section 13.9, Problem 24) and u is ergodic with respect to (JL (Section 13.9,
Problems 13 and 24). Show that for each e > 0 and each integer n > 0, there exists a
^-measurable subset A of X such that (i) the sets uJ(A) (0 gy <| n — 1) are pairwise

disjoint, and (ii) the complement of  (J uj(A) has measure g £ (Rokhlin's theorem).

(Choose a ^-measurable set B such that 0 < ju,(B) < e//i, and construct the correspon-
ding "Kakutani skyscraper" consisting of the sets up(Em) for m ^ 1 and 0 ^p < m
(Section 13.9, Problem 14(d)). Show that we may take A to be the union of the sets
) for all integers j > 0 and integers m such that m ^ (/ + ;

19. SUPPORT OF A MEASURE. MEASURES WITH COMPACT SUPPORT

(13.19.1)   If n is any measure on X, the union of all ju-negligible open sets is
\L-negligible (and hence is the largest ^-negligible open set).

For if U is this union, it follows from (1 3.1 .9) that the measure induced by
z on U is zero.

The complement in X of the largest /^-negligible open set is called the
support of n, and is denoted by Supp(^). To say that x e Supp(//) signifies that
lA*l*00>0 for every neighborhood V of x, or equivalently (13.5.1) that
M(l/l) > 0 for every /e Jfc(X) such that/(jc) ^ 0; or equivalently again,
that for each neighborhood V of x there exists a function /e Jf C(X) with
support contained in V, such that ju(/) ^ 0. If Supp(/x) = X, the only ^negli-
gible continuous function is the constant 0.

From the definition we have Supp(#) = Supp(|ju|), and it is clear that
Supp(^) = Supp(/i) for all scalars a ^ 0. More generally, if g is any locally
|ji|-integrable function, then we have Supp(# • //) c Supp(^) n Supp^u) ; for if
we put v = g • |/4 and if an open set U does not intersect Supp(#) or does not
intersect SuppOO, then |v|*(U) = 0 (13.14.1).

(13.19.2)   (i)   If [i and v are positive measures, then

Supp(/j + v) = SuppOu) u Supp(v).ists a continuous mapping TT of X onto I = [0,1]
```